Let $D=\{z \in \mathbb{C}|| z \mid<1\}$ be the open unit disk, and let $C$ be its boundary (the unit circle), with the anticlockwise orientation. Suppose $\phi: C \rightarrow \mathbb{C}$ is continuous. Stating clearly any theorems you use, show that

$$g_{\phi}(w)=\frac{1}{2 \pi i} \int_{C} \frac{\phi(z)}{z-w} d z$$

is an analytic function of $w$ for $w \in D$.

Now suppose $\phi$ is the restriction of a holomorphic function $F$ defined on some annulus $1-\epsilon<|z|<1+\epsilon$. Show that $g_{\phi}(w)$ is the restriction of a holomorphic function defined on the open disc $|w|<1+\epsilon$.

Let $f_{\phi}:[0,2 \pi] \rightarrow \mathbb{C}$ be defined by $f_{\phi}(\theta)=\phi\left(e^{i \theta}\right)$. Express the coefficients in the power series expansion of $g_{\phi}$ centered at 0 in terms of $f_{\phi}$.

Let $n \in \mathbb{Z}$. What is $g_{\phi}$ in the following cases?

1. $\phi(z)=z^{n}$.

2. $\phi(z)=\bar{z}^{n}$.

3. $\phi(z)=(\operatorname{Re} z)^{2}$.